Chapter 9: Q 41. (page 581)

Significance tests A test of $H_{o} : p = 0.5$ versus $H_{a} : p > 0.5$ based on
a sample of size $200$ yields the standardized test statistic $z = 2.19$ . Assume that the conditions for performing inference are met.
a. Find and interpret the P-value.
b. What conclusion would you make at the $α = 0.01$ significance level? Would
your conclusion change if you used α=0.05 instead? Explain your reasoning.
c. Determine the value of p^= the sample proportion of successes.

Short Answer

Expert verified

a. Hypothesis is rejected.

b. $α = 0.01 : fail to reject the null hypothesis.$ and $α = 0.05 : reject the null hypothesis$

c. $\hat{p} = 0.57774$

Step by step solution

Part (a) Step 1: Given Information

It is given that $z = 2.19$

$n = 200$

$α = 0.5$

$H_{0} : p = 0.5$

$H_{a} : p > 0.5$

Part (a) Step 2: Explanation

As $z = 2.19$

$P (x < z) = 0.98574$

$P (x > z) = 1 - 0.98574 = 0.014262$

When $P value \leq α$ , null hypothesis is rejected.

From above $P = 0.014262$

$P = 0.014262 < α$ , null hypothesis is rejected. $H_{0} : p = 0.5$

Part (b) Step 1: Given Information

It is given that $α = 0.01, α = 0.05$

Part (b) Step 2: Explanation

Higher the significance level, higher is probability to reject null hypothesis.

As $P = 0.014262 < 0.5$ , so null hypothesis is rejected.

If $α = 0.01$ $P = 0.014262 > 0.01$ , null hypothesis is not rejected.

If $α = 0.05$ , $P = 0.014262 > 0.05$ , null hypothesis is not rejected.

Therefore, $P$ value changes with change in significance levels.

Part (c) Step 1: Given Information

It is given that $n = 200$

$p = 0.5$

$z = 2.19$

Part (c) Step 2: Explanation

We know that $z = \frac{\hat{p} - p}{\sqrt{\frac{p (1 - p)}{n}}}$

Hence, $2.19 = \frac{\hat{p} - 0.5}{\sqrt{\frac{0.5 (1 - 0.5)}{200}}}$

$2.19 = \frac{\hat{p} - 0.5}{\sqrt{\frac{0.5 (0.5)}{200}}}$

$\hat{p} = (2.19 \times 0.03535) + 0.5$

Therefore, $\hat{p} = 0.57774$

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Short Answer

Step by step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Explanation

Part (b) Step 1: Given Information

Part (b) Step 2: Explanation

Part (c) Step 1: Given Information

Part (c) Step 2: Explanation

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